Areas [of the world] where [19th century] Protestant missionaries had a significant presence in the
past are on average more economically developed today, with
comparatively better health, lower infant mortality, lower corruption,
greater literacy, higher educational attainment (especially for women),
and more robust membership in nongovernmental associations.

As far as he can tell, the presence of such missionaries was far more significant than any other factor(s) (culture, climate, other historical factors etc.) in making this difference. He doesn't know exactly what caused this difference, but he notes that missionaries promoted the idea that all people were of equal worth and also that they tended to teach reading and establish both schools and printing presses - all things that could tend towards the establishment of democracy.

Even in places where very few people converted, missionaries seem to have had a profound economic and political impact. However, independence from the colonial state appeared to be necessary. Catholic missionaries often were state-sponsored. Woodberry notes:

One of the main
stereotypes about missions is that they were closely connected to
colonialism. But Protestant missionaries not funded by
the state were regularly very critical of colonialism.

I know that missionaries here in New Zealand were influential in the writing of the Treaty of Waitangi and were also energetic in protesting various government and settler land grabs.

Although the article I read is from a Christian publication, Woodberry's work has apparently been published in a prestigious, peer-reviewed sociological journal. It's a single study, so there could well be another explanation of this effect that he has missed, but his secular academic peers are taking the claims seriously.

The article concludes with 8 brief biographies of 19th century protestant missionaries who made a significant difference in the countries in which they worked.